๐ Topic Summary
Exponential equations involve variables in the exponent, such as $2^x = 8$. To solve these equations, especially when you can't easily express both sides with the same base, we use logarithms. A logarithm is the inverse operation to exponentiation. If $a^x = b$, then $log_a(b) = x$. By taking the logarithm of both sides of an exponential equation, we can bring the exponent down and solve for the variable. The most common logarithms are the common logarithm (base 10) and the natural logarithm (base $e$).
For example, to solve $3^x = 15$, take the logarithm of both sides: $log(3^x) = log(15)$. Using the power rule of logarithms, $x \cdot log(3) = log(15)$. Finally, divide by $log(3)$ to find $x = \frac{log(15)}{log(3)}$. Approximating, $x \approx 2.465$.
๐ง Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Exponential Equation |
A. The inverse operation to exponentiation. |
| 2. Logarithm |
B. The power to which a base must be raised to obtain a number. |
| 3. Base |
C. An equation where the variable is in the exponent. |
| 4. Argument |
D. The number inside the logarithm. |
| 5. Power Rule |
E. $log_b(x^p) = p \cdot log_b(x)$ |
โ๏ธ Part B: Fill in the Blanks
To solve exponential equations, we often use __________. A logarithm is the __________ operation of exponentiation. The common logarithm has a base of __________, while the natural logarithm has a base of __________. Using the __________ rule of logarithms helps simplify equations by bringing the exponent down.
๐ค Part C: Critical Thinking
Explain in your own words why logarithms are useful for solving exponential equations. Give an example of a real-world situation where solving exponential equations might be necessary.